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A full interpretation of this remark will require us to pick up the precise technical sense in which Peirce is using the word "collection", and that will take us back to his logical reconstruction of certain aspects of number theory, all of which I am putting off to another time, but it is still possible to get a rough sense of what he's saying relative to the present frame of discussion.

A full interpretation of this remark will require us to pick up the precise technical sense in which Peirce is using the word ''collection'', and that will take us back to his logical reconstruction of certain aspects of number theory, all of which I am putting off to another time, but it is still possible to get a rough sense of what he's saying relative to the present frame of discussion.

The "number of" map ''v'' : ''S'' → '''R''' evidently induces some sort of morphism with respect to logical sums.  If this were straightforwardly true, we could write:

The "number of" map <math>v : S \to \mathbb{R}</math> evidently induces some sort of morphism with respect to logical sums.  If this were straightforwardly true, we could write:

: (?) ''v''(''x'' +, ''y'') = ''vx'' + ''vy''

v(x ~+\!\!,~ y) & = & vx ~+~ vy & ?

Equivalently:

Equivalently:

: (?) [''x'' +, ''y''] = [''x''] + [''y'']

[x ~+\!\!,~ y] & = & [x] ~+~ [y] & ?

Of course, things are just not that simple in the case of inclusive disjunction and set-theoretic unions, so we'd "probably" invent a word like "sub-additive" to describe the principle that does hold here, namely:

Of course, things are just not that simple in the case of inclusive disjunction and set-theoretic unions, so we'd probably invent a word like ''sub-additive'' to describe the principle that does hold here, namely:

: ''v''(''x'' +, ''y'') =< ''vx'' + ''vy''

v(x ~+\!\!,~ y) & \le & vx ~+~ vy

Equivalently:

Equivalently:

: [''x'' +, ''y''] =< [''x''] + [''y'']

[x ~+\!\!,~ y] & \le & [x] ~+~ [y]

This is why Peirce trims his discussion of this point with the following hedge:

This is why Peirce trims his discussion of this point with the following hedge:

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Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves — provided all the terms summed are mutually exclusive. (CP 3.67).

<p>Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves &mdash; provided all the terms summed are mutually exclusive.</p>

 

<p>(Peirce, CP 3.67).</p>

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